In this paper, a family of Steffensen type methods of fourth-order convergence for solving nonlinear smooth equations is suggested. In the proposed methods, a linear combination of divided diferences is used to get a better approximation to the derivative of the given function. Each derivative-free member of the family requires only three evaluations of the given function per iteration. Therefo...

In this paper, a family of Steffensen-type methods of optimal order of convergence with two parameters is constructed by direct Newtonian interpolation. It satisfies the conjecture proposed by Kung and Traub (J. Assoc. Comput. Math. 1974, 21, 634–651) that an iterative method based on m evaluations per iteration without memory would arrive at the optimal convergence of order 2m−1. Furthermore, ...

The problem is to calculate a simple zero of a non-linear function f. We consider rational iterations without memory which use two evaluations of f or its derivatives. It is shown that the optimal order is 2. This settles a conjecture of Kung and Traub that an iteration using n evaluations without memory is of order at most 2 "^, for the case n = 2. Furthermore we show that any rational two-eva...

A class of three-step eighth-order root solvers is constructed in this study. Our aim is fulfilled by using an interpolatory rational function in the third step of a three-step cycle. Each method of the class reaches the optimal efficiency index according to the Kung-Traub conjecture concerning multipoint iterative methods without memory. Moreover, the class is free from derivative calculation ...

In the study of systems’ dynamics presence symmetry dramatically reduces complexity, while in chemistry, plays a central role analysis structure, bonding, and spectroscopy molecules. more general context, principle equivalence, local symmetry, dictated gravity, space-time itself. certain instances, especially we end up having to deal with an equation multiple roots. A variety optimal methods ha...

In this paper, we present a fourth order method for computing simple roots of nonlinear equations by using suitable Taylor and weight function approximation. The method is based on Weerakoon-Fernando method [S. Weerakoon, G.I. Fernando, A variant of Newton's method with third-order convergence, Appl. Math. Lett. 17 (2000) 87-93]. The method is optimal, as it needs three evaluations per iterate,...

A general class of three-point iterative methods for solving nonlinear equations is constructed. Its order of convergence reaches eight with only four function evaluations per iteration, which means that the proposed methods possess as high as possible computational efficiency in the sense of the Kung-Traub hypothesis (1974). Numerical examples are included to demonstrate a spectacular converge...

In this paper we will examine self-accelerating in terms of convergence speed and the corresponding index of efficiency in the sense of Ostrowski–Traub of certain standard and most commonly used in practice multipoint iterative methods using several initial approximations for numerical solution of nonlinear equations due to optimal in the sense of the Kung–Traub algorithm of order 4, 8 and 16. ...